Now writing , we obtain

(10.19)

From the definition of , we have

and

(10.20)

We shall show that

for

Assume that

for some

Since for small positive values of , we may assume without loss of generality that

(10.21)

By the mean value theorem,

for some

or for

it follows that

(because of the assumption that .

Applying the mean value theorem to the function , we have

for some . In view of we then have (because ). This contradicts the differential equation (10.19) in view of and (10.21). It thus follows that . The desired relation (10.18) is the special ease .